Measurement and determination of Stokes parameters

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The interstellar medium

The interstellar medium (ISM) is the material that, in a galaxy, fills the space between stars and blends into the surrounding intergalactic environment. Its density is of the order of a few particles per cm3 . Despite this, the interstellar medium in our Galaxy accounts for about 10% of its total mass. The matter of the interstellar medium is in two forms:
• Gas, of which 89% is hydrogen, 9% helium and 2% heavy elements (carbon, oxygen, nitrogen, etc.).
• Dust, which represents 1% of the total mass of the ISM.
It is present in all spiral galaxies, barred spiral, irregular and almost non-existent in elliptical and lenticular galaxies. The interstellar medium bathes in a field of radiation produced by the stars present in the galaxies. The inter-stellar medium is composed of several phases that are distinguished by their temperature, their density and the state of matter. The different components of the ISM are presented in table 1.1.

The Galactic magnetic field

The presence of an interstellar Galactic magnetic field was postulated for the first time in 1937 by Hannes Alfv´en and then by Enrico Fermi in 1949. Its presence was later confirmed by the observation in 1949 of the linear polar-ization of light emitted by nearby stars and the observation of synchrotron emission in the Milky Way in 1950. The intensity of the Galactic magnetic field is of the order of a few micro-gauss (about 10,000 times weaker than the Earth’s magnetic field). It plays an important role in many astrophys-ical processes. For example, the magnetic field opposes the gravitational force to explain the thickness of the Galactic disk. It is also essential in the stellar formation process, initially limiting the collapse of the molecular clouds by opposing the gravitation and then allowing the continuation of the collapse process by allowing to evacuate kinetic momentum to the outside of the stellar system. Finally, during magnetic reconnection and ambipolar diffusion processes, a large amount of energy is dissipated mainly in ther-mal form which constitutes an important source of heating of the ISM. The Galactic magnetic field is generated and driven by a Galactic dynamo oper-ating throughout the entire volume of the Galaxy under the combined action of its rotation and the turbulent movements generated by the explosion of supernovae.

Interstellar dust

By crossing the clouds of the interstellar medium, the light emitted by the stars is attenuated. It is called the ”extinction” which is due to the absorp-tion and the diffusion, by the dust grains, of the radiation. Figure 1.9 shows the normalized interstellar extinction curves from the far-IR through the UV. This curve provides a basis for making hypotheses about the characteristics of the interstellar medium grains. Based on these observations, we suppose that dust of the interstellar medium can be classified into three main families:
• The very small grains whose sizes vary between 0.4 nm and 1.2 nm. Responsible for the characteristics of the extinction curve in the far-UV.
• Big grains with an average size of 100 nm composed of ice-coated graphites and silicates. These grains are mainly responsible for the extinction in visible and infrared and partially responsible for the ex-tinction in near UV.
• PAHs (Polycyclic Aromatic Hydrocarbons), molecules consisting of a few tens of carbon and hydrogen atoms organized in aromatic cycles. The presence of these explain the UV bump around 217.5 nm.
The constituents of dust (silicate and graphite) are produced by stars. The stars form in the interstellar medium by gravitational collapse of cold and dense cores (or clouds). Most of the life of a star happens on the main sequence defined by the Hertzsprung-Russell diagram relating the mass of a star on the one hand, the radius and luminosity of this star on the other. The density of the surrounding environment of stars is large enough to allow the formation of more complex molecules and for the formation of dust. At the end of their life, when they emerge from the main sequence, the stars eject matter in the form of stellar winds or supernova explosion for the most massive ones. This material is the main constituent elements of the dust of the interstellar medium. These dust grains will then mix, with the gas present in the MIS, to form an interstellar cloud which, by collapse, will again permit the formation of a star. These stages constitute what is called the cycle of matter or dust cycle in the interstellar medium (see figure 1.10).
In the Milky Way, dust is distributed mainly in the galactic plane with a higher concentration around the central nucleus. When moving away from the plane, the dust density becomes much lower (see figure 1.11).

Observations of polarized emission from dust

The first observation of the dust polarized emission at 850 µm (353 GHz) over a large fraction of the Galactic plane was made by the Archeops balloon mission (Benoit et al. 2004). These measurements indicated high polarization levels (up to 15%) in the diffuse ISM. In continuity with Archeops, the Planck satellite, launched in 2009, has produced the first all-sky map of the polarized emission from dust at sub-mm wavelenghts (Planck Collaboration. XI. et al. 2014). This survey was an immense step forward in sensitivity, coverage and statistics on the main polarization parameters. Planck provided new insight into the structure of the galactic magnetic field and dust properties. It also provided the first statistical characterization of the main foreground to CMB. I will try to summarize here some important results provided by the mission.
The link between the hydrogen column density (denoted NH ) and the po-larization fraction pl has been studied (Planck Collaboration Int. XX. et al. 2015). It shows that the maximum value of pl is high (around 20%) and that it has been observed in regions with a moderate column density (less than 2 · 1021cm−2). In addition, a statistical analysis has shown that p is decreasing with NH above 1021cm −2, and is anti-correlated with the angle dispersion function characterizing the spatial structure of the polarisation angle. The polarization angle is ordered over extended areas that are sepa-rated by structures where the sky polarisation changes abruptly. Comparison of this observation with magnetohydrodynamic simulations tends to attribute these scatter of p to fluctuations in the orientation of the magnetic field lines along the line of sight.
The study of the role of the magnetic field in the formation of interstellar medium structures has also been studied (Planck Collaboration Int. XXX et al. 2016). The Planck intensity map shows elongated structures (filaments or ridges) also visible on the polarization maps. These structures seem pref-erentially aligned with the magnetic field especially. Toward denser regions, the relative orientation changes progressively from parallel in areas with low-est hydrogen column density to perpendicular in areas with highest hydrogen column density. Simulations show that such a change could be related to the degree of magnetisation of the cloud, particularly when magnetic energy is in equipartition with turbulent energy (Hennebelle 2013, Soler et al. 2013, Chen et al. 2016). As previously explained, polarized dust emission from the ISM represents the main foreground of CMB polarization measurements above 100 GHz. The polarized dust angular power spectra ClBB et ClEE were measured by Planck on all multipoles ` between 40 and 600 far away from the Galactic plane (Planck Collaboration Int. XXX et al. 2016). The polarization power spectra of the dust are well described by power laws in multipole, Cl ∝ lα with α = −2.42 ± 0.02 for both EE and BB spectra. The data has also shown that there is no region of the sky for which CMB B-mode polariza-tion measurements can be made without having to previously remove the polarized dust emission.

PILOT scientific objectives

PILOT (for Polarized Instrument for the Long-wavelength Observation of the tenuous ISM) is an astrophysical experiment designed to measure the polarized emission of light by interstellar medium dust in the far infrared (FIR). The band used for observations with PILOT is centered around 240 µm. The experiment is designed to fly under a stratospheric balloon at an altitude of 40 km, in the stratosphere. This allows us to limit the absorption and the thermal noise produced by the atmosphere. The PILOT experiment is embarked under a stratospheric balloon, with a volume of approximately 800 000 m3 to reach this altitude during flights. Figure 1.14 shows a view of the PILOT gondola under the Australian sky during the second campain in Alice Springs. The scientific objectives of the mission are multiple. One of the objectives is to use the polarized emission of the dust grains of the in-terstellar medium to map the direction and intensity of the magnetic field of our Galaxy. These measurements will also lead to a better understanding of the magnetic properties of the grains of the interstellar medium. The PILOT measurements are complementary to those carried out by the Planck satel-lite. They are made at a wavelength closer to the maximum light emission of the interstellar dust. This allows us to obtain more flux. This also has the advantage of providing a better angular resolution for a given mirror size, the angular resolution of an instrument being proportional to the ratio of the observed wavelength to the mirror diameter (θ = 1.22 ∗ λ/D). The other major objective of the mission is to use dust-polarized emission measure-ments on diffuse regions of the sky with unprecedented sensitivity. PILOT will play a crucial role in the preparation of missions for measuring the po-larization of the cosmic diffuse background to help in the characterisation of the foreground polarisation. The PILOT data will help us to constrain the variations of polarization fraction as a function of wavelength in the infrared range which is an important aspect of extrapolating dust polarization data obtained at high frequencies to CMB wavelengths.

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Internal Calibration Source (ICS)

An internal calibration source (ICS), placed behind the mirror M3 (see fig-ure 2.8), is used to allow an intercalibration of the bolometers in the most precise way possible to facilitate the reconstruction of the polarization mea-surements. The source used is the spare model of the SPIRE instrument on the Herschel satellite. The ICS was used during each scene to calibrate the detector response variations. The ICS is controlled in current with a square modulation and a maximum current of 2 mA. The voltage and current are measured continuously in order to trace the variations of the electric power dissipated in the source independently of the changes of its impedance related to the variations of temperatures. The ICS was developed by the University of Cardiff, Great Britain.

House-Keeping Electronics

A dedicated electronics (UGTI, for ”Unit´e de gestion des donn´ees Techniques de l’Instrument”) is used for a variety of house-keeping tasks. In particular, the UGTI monitors cryogenic temperatures in the range 2-77K inside the cryostat, and ambient temperatures for the rest of the instrument. The UGTI is also used to regulate the intensity of the current inside the ICS calibration source, to operate cryogenic valves that open and close the cryostat helium tank, and to operate heaters on the outer shell of the cryostat and in the helium tank.

Stellar Sensor

PILOT has a stellar sensor called Estadius developed by CNES. It is equipped with a gyroscopic optic fiber, a 16 megapixel CCD camera and a large aper-ture lens with a focal length of 135mm. The field of view of the stellar sensor is 10ox15o. The system provides accurate star position measurements due to the small angular size of each pixel, and is able to detect stars against a bright sky background. It allows us a reconstruction of the pointing in the order of 1 arcsec in translation and 6 arcsec in rotation at 1σ. Estadius demonstrates good autonomy through the use of automatic constellation recognition. A schematic view of Estadius is visible in figure 2.9. A complete description of the stellar sensor is presented in Montel (2015).

Polarization measurement with PILOT

The signal measured on the sky by the instrument is defined according to the Stokes parameters as follows: m = 1 hRxyTxy · [I ± Q cos(4ω + 2ϕ) ± U sin(4ω + 2ϕ)] + Oxyi (2.1).
with I the total intensity, Rxy the detector response, Txy is the optics transmission. The ± sign is positive (+) for the TRANS arrays and negative (−) for the REFLEX arrays. The additional term Oxy is to account for an arbitrary electrical or signal offset. ω and ϕ are respectively the Half-Wave plate angle and the parallactic angle defined as described in chapter 8. The calculations linking the Stokes parameters to the measurement carried out by PILOT are detailed in chapter 8.
The measurement of the polarization state is carried out using the half-wave plate and the fixed grid polarizer (see figure 2.10). They make it possible to modify the state of polarization of the incident beam and to separate it into two beams of orthogonal polarization state. In the context of PI-LOT observations, the half-wave plate remains in a fixed position during the same observation, in contrast to other systems where the HWP rotates con-tinuously during an observation. Each observation must be carried out at minimum with two different HWP positions in order to optimally reconstruct the Stockes parameters I, Q, U allowing us to describe the polarization state of a wave (see chapter 8). When an object is observed with a certain posi-tion of the half wave plate, we call this observation a ”scene”. A scene can be subdivided into different sequences:
• SLEW: the slews correspond to the passage between the end of a scene and the start of the next scene. Slews are performed simultaneously in azimuth and elevation. HWP position changes occur at the end of the Half-Wave Plate and the se-lection of two orthogonal states by the fixed grid polarizer. Figure adapted from Engel (2012)
• SCAN: scans correspond to a sweep of a portion of a scene (at constant or variable elevation. see section and ).
• CALIB: these sequences correspond to the time when the internal cal-ibration source is functioning. These sequences are performed between two scans.
• CHAZEL: This corresponds to relative displacements of the pointed load in azimuth and elevation. These are the transition movements from one scan to another.

Table of contents :

I Scientific context and PILOT instrument 
1 Scientific Context 
1.1 Infrared astronomy and polarization of light
1.1.1 Thermal emission
1.1.2 Polarization of an electromagnetic wave
1.2 Cosmic Microwave Background (CMB)
1.2.1 The Big Bang
1.2.2 Polarization of the CMB
1.2.3 Observational bias
1.3 The interstellar medium
1.3.1 The Galactic magnetic field
1.3.2 Interstellar dust
1.3.3 Thermal emission of dust
1.3.4 Polarized emission and extinction of galactic dust
1.4 Observations of polarized emission from dust
1.5 PILOT scientific objectives
2 PILOT instrument 
2.1 Description of the instrument
2.1.1 The gondola
2.1.2 Optics
2.1.3 The Half Wave Plate
2.1.4 Detectors
2.1.5 The cryostat
2.1.6 Filters
2.1.7 Internal Calibration Source (ICS)
2.1.8 House-Keeping Electronics
2.1.9 Stellar Sensor
2.2 Polarization measurement with PILOT
2.3 PILOT data format
2.3.1 Pilot Instrument MOdel (PIMO)
2.3.2 Data File Format
2.4 PILOT flights
2.4.1 First Flight
2.4.2 Second Flight
II Inflight performances PILOT 
3 Time constants 
3.1 Data-reading time delay
3.2 Time constant of the ICS
3.3 ICS ground tests
3.4 Glitches in-flight measurments.
3.4.1 Glitches detection method.
3.4.2 Time constant measurement.
3.5 ICS in-flight measurments
4 Pointing reconstruction 
4.1 Stellar sensor
4.2 Focal plane geometry
4.3 Sky Coordinates Calculations
4.3.1 Coordinate systems
4.3.2 Euler angles
4.3.3 Quaternions
4.3.4 From Euler Angles to quaternions.
4.3.5 Coordinates calculations
4.4 Issues in pointing reconstruction
4.5 Compact sources method
4.6 Maximum correlation method
4.7 The barycentric method
4.8 Preliminary pointing model
4.9 Coordinates versions
5 Noise properties 
5.1 Noise sources
5.1.1 Photon noise
5.1.2 Thermal noise
5.1.3 Johnson-Nyquist noise
5.1.4 Flicker noise
5.1.5 Readout noise
5.1.6 Environmental noise
5.2 Ground calibrations
5.3 Calculation of noise levels and power spectra
5.4 Noise levels and spatial distribution
5.5 Noise power spectra
5.6 Half-pixel difference noise power spectra
5.7 Temperature fluctuations at 300 mK
6 Response and Background of the detectors 
6.1 Background
6.1.1 Backgroung level
6.1.2 Background polarization
6.2 Responses
6.2.1 Temporal variations
6.2.2 Response Variations with Background
7 Optical quality 
7.1 Ground tests
7.2 In-flight PSF
7.3 Simulated PSF
III Map making 
8 Polarisation measurement 
8.1 Stokes parameters
8.2 Angles definition
8.3 Mueller matrices
8.4 Measurement and determination of Stokes parameters
9 Map-Making and preliminary results 
9.1 Map-making algorithms
9.1.1 Scanamorphos
9.1.2 ROMA
9.2 Data processing pipeline
9.2.1 Time constant deconvolution.
9.2.2 Deglitching
9.2.3 Atmospheric subtraction
9.2.4 Responses correction
9.3 Simulations
9.4 Preliminary results
9.4.1 I,Q,U maps
9.4.2 Polarisation angles
9.5 Work in progress.
Conclusions and perspectives (english version)
Conclusions et perspectives (version fran¸caise)
A Experimental Astronomy paper 
B SPIE conference paper


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